Page 262 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 8.9 Cholesky Decomposition 249

By letting A = co^m/k, the equation of motion is then reduced to

1.50 -0.707 - A
-0.707 1
and its characteristic equation becomes
(1.50-A ) -0.707

= 0 and 2.50A -f 1 = 0
-0.707 (1 -A )
The eigenvalues and eigenvectors solved from these equations are
fO.707)
A, = 0.50
^2/ ^ 1 1-000
\ ^2) I -
A, = 2.00
^^2/ 1-000^
These are the modes in the y coordinates, and to obtain the normal nodes in the
original x coordinates, we first assemble the previous modes into a modal matrix Y
from which the modal matrix in the x coordinates is found.
0.707 1.414
y = 1.000 1.000
0.707 0 0.707 -1.414 0.50 -1.00
0 1.0 1.00 1.00 1.00 1.00
These results are in agreement with those found in Example 6.8-2.

8.9 CHOLESKY DECOMPOSITION

When matrix M or K is full, matrices U and can be found from the Cholesky
decomposition. In this evaluation, we simply write the equation M = U^U (or
K = U^U) in terms of the upper triangular matrix for U and its transpose as
U — M
n 0 0 «12 ^13 mil m,2 mi3
0 0 «22 ^23 = mix W22 ^^23 (8.9-1)
0 0 ^33 m^x "Í32 ^^33
By multiplying out the left side and equating to the corresponding terms on the
right, each of the u¿j can be determined in terms of the coefficients on the right
side.
The inverse of the upper triangular matrix is determined from the equation
- /:
U u-^ I
“ 1 0 o'
1^11 11 b\2 ^13
0 ) bj. b.. 0 1 0 (8.9-2)
/22
0 0 0 0 0 1